Download LOVELY PAIRS OF MODELS: THE NON FIRST ORDER CASE

LOVELY PAIRS OF MODELS: THE NON FIRST ORDER CASE
ITAY BEN-YAACOV
Abstract. We prove that for every simple theory T (or even simple thick compact
abstract theory) there is a (unique) compact abstract theory T P whose saturated
models are the lovely pairs of T . Independence-theoretic results that were proved in
[BPV03] when T P is a first order theory are proved for the general case: in particular
T P is simple and we characterise independence.
Introduction
Lovely pairs of models of a simple first order theory were defined in [BPV03]. Under
an additional assumption, namely that the equivalent conditions of Fact 0.2 below hold,
it is shown that lovely pairs provide an elegant means for the study of independencerelated phenomena in such a theory. This generalises a similar treatment of stable
theories through the study of beautiful pairs in [Poi83].
A lovely pair of models of T is given by (M, P ) where M ² T and P is a new
unitary predicate defining an elementary sub-structure with quite a few additional
properties (see Definition 2.1 below). The following is proved in [BPV03] (the analogue
for beautiful pairs of models of a stable theory is proved in [Poi83]):
Fact 0.1. Let T be a complete simple first order theory. Then all lovely pairs of T
have the same first order theory T P in the language L ∪ {P }.
This does not mean, however, that the complete first order theory T P is meaningful.
For example, in order to use T P for the study of lovely pairs we would like saturated
models of T P to be ones. In fact, it is proved that:
Fact 0.2. The following conditions are equivalent (for a first order simple theory T ):
(i) The |T |+ -saturated models of T P are precisely the lovely pairs.
(ii) There is a |T |+ -saturated model of T P which is a lovely pair.
(iii) The notion of elementary extension of models of T P coincides with that of a
free extension (Definition 1.2).
(iv) Every model of T P embeds elementarily in a lovely pair.
Date: March 12, 2004.
2000 Mathematics Subject Classification. 03C45,03C95.
Key words and phrases. simple theories – lovely pairs.
This is the result of research conducted in the University of Illinois at Urbana-Champaign, as a part
of CNRS-UIUC collaboration. The author would like to thank Anand Pillay and Evgueni Vassiliev
for hospitality and fruitful discussions.
At the time of the writing of this paper, the author was a graduate student with the Équipe de
Logique Mathématique of Université Paris VII.
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If this holds (the “good” case), then T P is simple as well, and provides an elegant
means for the study of certain independence-related properties of T itself, as mentioned
above. If this fails (the “bad” case), then the first order theory T P is pretty much
useless. This was noticed by Poizat in the stable case, where things go well if and only
if T does not have the finite cover property; the analogous criterion for simple theories
can be argued to be the correct analogue of non-f.c.p. in simple theories.
The goal of the present article is to show that in the proper context, one can do in
the “bad” case just the same things as in the “good” one. By the previous discussion
it should be clear that this cannot be done in first order model theory, and we need
to look for a more general framework. Such a framework, that of compact abstract
theories, or cats, is exposed in [Ben03a]. Simplicity theory is developed for cats in
[Ben03b], but has a few setbacks with respect to first order simplicity. In [Bena] we
define the notion of a thick cat (which is still much more general than a first order
theory), and prove all basic properties of simplicity theory in this framework.
Here we prove that if T is a thick simple cat (so in particular, if T is a simple first
order theory), then there exists a unique cat T P, whose saturated models are precisely
the lovely pairs of T . T P is also thick and simple and has a language of the same cardinality as T . In addition, there is a description (a notion close to interpretation, defined
in [Bena]) of T P in T , which gives us an elegant characterisation of independence in
T P. We also prove that if T is Hausdorff, semi-Hausdorff, supersimple, stable, stable
and Robinson, or one-based, then so is T P.
It follows from Proposition 2.3 below that T P is (equivalent to) a first order theory
if and only if T is and the equivalent conditions of Fact 0.2 hold. Thus, for a first
order theory T , the “good” and “bad” cases are simply the first order case and the
non-first-order one, respectively, of which the former was studied in [BPV03]. In the
present paper, however, such considerations as whether T P is first order or not are
hardly of any importance.
The fundamental tool is the construction of a cat from a compact abstract elementary
category, as described in [Ben03a]. This tool allows us in certain cases to fix the notion
of elementary extension as we like: since we know that things go well if and only if
the elementary extensions are the free extensions, we turn things around and try to
construct a cat where free extensions play the role of elementary ones. As it turns out,
this is indeed one of the cases where this technique works, and the only assumption on
the original theory we actually use is that it is a thick simple cat.
We can think of several reasons why this may be an interesting thing to do: First,
this gives a nice and rather comprehensive set of examples of the basic tools used in the
framework of cats, and in particular of simplicity theory. Second, this is an additional
example supporting our thesis that simplicity in thick cats lacks nothing in comparison
with simplicity in first order theories (alas, this is not true for simplicity in arbitrary
cats). Third, and maybe most important, thick simple cats are (or at least, seem to be)
the correct framework for the treatment of lovely pairs, and therefore results proved
in this context should be the most general.
Moreover, this framework allows us to state and prove results that are either unnatural or altogether meaningless in the first order case. Even when proving something that
makes perfect sense in a first order theory, we may use for its proof tools that would
be unnatural in the treatment of a first order theory, and this may eventually yield
LOVELY PAIRS OF MODELS: THE NON FIRST ORDER CASE
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a simpler or more elegant proof. In fact, some results appearing in [BPV03] (notably
the preservation of one-basedness) were originally proved quite easily in this context,
and it took a bit of effort to find first order counterparts for the “feline” proofs.
Let us give a few reminders concerning cats. Most of this comes from [Ben03a].
Definition 0.3. Let L be a first order language, and fix a positive fragment of L,
i.e., a subset ∆ ⊆ L which is closed for positive boolean combinations (actually, L is
completely unimportant, all we want is ∆). A formula, unless otherwise qualified, is
always a member of ∆, and similarly for partial types.
A universal domain (with respect to ∆) is a structure U satisfying:
(i) Strong homogeneity: If A, B ⊆ U are small and f : A → B is a ∆homomorphism (i.e., for every ϕ ∈ ∆ and a ∈ A, U ² ϕ(a) =⇒ U ² ϕ(f (a))),
then f extends to an automorphism of U (so in particular, f is a ∆isomorphism of A and B).
(ii) Compactness: Every small partial ∆-type over U which is finitely realised in
U is realised in U .
Although this is not required by the definition, we will also assume that every existential
formula, i.e., formula of the form ∃y ϕ(x, y) where ϕ ∈ ∆, is equivalent in U to a partial
∆-type. (If not, we can always close ∆ under existential quantification without harming
either compactness or homogeneity; this is just usually unnecessary.)
Saturated and strongly homogeneous models of first order theories are one example
of universal domain (with ∆ = L). Another easy example which we will refer to later
on is that of Hilbert spaces:
Example 0.4. Let H be the unit ballP
of a very large Hilbert space. Let ∆ be the
set of all formulas of the form s ≤ k i<n λi xi k ≤ r (closed under positive boolean
combinations). Then H is a universal domain w.r.t. ∆.
The negative universal theory of a universal domain
ThΠ (U ) = {∀x̄ ¬ϕ : ϕ(x̄) ∈ ∆, U ² ∀x̄ ¬ϕ(x̄)}
has the property that the category of subsets of its e.c. models has the amalgamation
property (there is a little twist here, since the notion of e.c. models is defined with
respect to ∆-homomorphisms). A negative universal theory having this property is
called a positive Robinson theory. Conversely, if T is a positive Robinson theory, and
in addition is complete (i.e., the category of its e.c. models has the joint embedding
property), then T = ThΠ (U ) for some universal domain U ; otherwise, every completion
of T has a universal domain. Thus the giving of a universal domain is essentially the
same as the giving of a complete positive Robinson theory. Henceforth, a theory means
a positive Robinson theory, unless explicitly stated otherwise.
To a universal domain U , or to a theory T , we associate type-spaces: for every set
of indices I we define SI (T ) as the set of all maximal types in α variables which are
consistent with T . If U is a universal domain for T then this is the same as U I / Aut(U ),
by homogeneity. We put a compact and T1 topology on SI (T ) by taking the closed sets
to be those defined by partial types. If Sn (T ) is Hausdorff for every n < ω then SI (T )
is Hausdorff for every set I, and we say that T is Hausdorff. One consequence of being
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Hausdorff is that the property of two tuples to have the same type is a type-definable
property. If only the latter holds, we say that T is semi-Hausdorff. An even weaker
property is thickness, defined in [Bena], which says that indiscernibility of sequences
is type-definable.
We render the mapping I 7→ SI (T ) a contravariant functor in the obvious manner:
if f : I → J is any mapping, then f ∗ : tp(aj : j ∈ J) 7→ tp(af (i) : i ∈ I) defines a
continuous mapping f ∗ : SJ (T ) → SI (T ). We call this the type-space functor of T ,
denoted S(T ). Conversely, up to a change of language, we can reconstruct the positive
Robinson theory T from S(T ) (see [Ben03a, Theorem 2.23]).
Finally, in [Ben03a, Section 2.3] we characterise when a class of structures equipped
with a notion of embedding has a universal domain which is also a universal domain
for a positive Robinson theory. First, we represent such a class with concrete category
M all of whose morphisms are injective (the embeddings); we call it an elementary
category with amalgamation if it satisfies some additional properties: Tarski-Vaught,
elementary chain and amalgamation (see [Ben03a, Definition 2.27]). In particular,
amalgamation gives us a reasonably good notion of type: if M and N are models (i.e.,
objects of M) and ā ∈ M and b̄ ∈ N are tuples of the same length, then they have
the same type if and only if we can embed M and N in a third model P such that the
images of ā and b̄ in P coincide. This defines a contravariant functor S(M) from sets
to sets as above. Using this notion of types we obtain some rudimentary semantics
that allow us to state the three last requirements (see [Ben03a, Definition 2.32]): that
the collection of types is not a proper class; that types of infinite tuples are determined
by the types of finite sub-tuples; and most importantly, that we can put compact and
T1 topologies on each SI (M) such that its morphisms are closed continuous mappings.
This last requirement is equivalent to saying that there is a language L, a positive
fragment ∆ ⊆ L, and a way to render every object of M an L-structure, such that:
(i) The satisfaction of a ∆-formula by a tuple in M ∈ M is determine by the type
of the tuple.
(ii) If Σ is a set of ∆-formulas, possibly in infinitely many variables, and Σ is
finitely realised in M, then it is realised in M.
If all these requirements hold, then there exists a positive Robinson theory T (in fact
T = ThΠ (M)) satisfying S(T ) ∼
= S(M), and e.c. models of T embed in models of
M and vice versa. Also, T is complete if and only if M has the joint embedding
property, and in this case a universal domain for T is a universal domain for M in
some reasonable sense.
Since we have three equivalent presentation (a positive Robinson theory, a compact
type-space functor and a compact elementary category) of the same concept, we prefer
to refer to this concept with a generic name: compact abstract theory, or cat. The third
presentation of cats is the main tool we use in the first section. In the fourth section
we concentrate on the second approach and study the relations between the type-space
functors of our theory T and of the theory of its pairs T P.
As for simplicity and independence, the thumb rule is that everything that’s true in a
simple first order theory (by which we mean the main results of [Kim98, KP97, HKP00])
is true in a simple thick cat. Part of this is shown for arbitrary simple cats in [Ben03b]
LOVELY PAIRS OF MODELS: THE NON FIRST ORDER CASE
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and the rest (in particular the extension axiom) is shown in [Bena] under the hypothesis
of thickness.
We consider the distinction between the “real” sorts and the hyperimaginary sorts
immaterial: an element is usually a real one, but as we may adjoin any hyperimaginary
sort to the original theory it may in fact be in any such sort. We use lowercase letters to
denote elements and (possibly infinite) tuples thereof, and uppercase letters to denote
sets of such elements or tuples (of course, any set can be enumerated into a tuple, but
sometimes it’s convenient to make a conceptual distinction).
We recall that if a is a tuple of elements, or even a hyperimaginary element, then
bdd(a) (respectively dcl(a)) is the collection of all hyperimaginary elements b such that
tp(b/a) has boundedly many realisations (respectively, a unique realisation). If A ⊆ B
then A is boundedly closed in B if B ∩ bdd(A) = A.
It is also a fact that if c ∈ bdd(b) then a ^
| b c for every a, and a ^
| b a if and only if
a ∈ bdd(b).
1. The category of T -pairs
Convention 1.1. We fix a thick simple cat T .
We may consider it as a positive Robinson theory with respect to a positive fragment ∆.
By an elementary mapping we mean a ∆-elementary one, that is a ∆-homomorphism.
We do not assume that T is complete. Therefore, instead of working inside a single
universal domain for T , we work with the category of e.c. models of T (or more precisely, of subsets thereof). The reader should keep in mind the existence of a partial
elementary mapping between two e.c. models of T implies that they are models of
the same completion, so we could assume that T is complete without much loss of
generality.
We aim at the construction of T P. Our starting point is the notions of pair and free
extension/embedding:
Definition 1.2.
(i) A pair is a couple (A, P ) where A is a subset of some e.c.
model of T , and P is a unary predicate on A, such that P (A) is boundedly
closed in A (i.e., A ∩ bdd(P (A)) = P (A)). We allow ourselves to omit P when
no ambiguity may arise, convening that it is part of the structure on A.
(ii) A free embedding of pairs f : (A, P ) → (B, P ) is an elementary embedding f :
A → B such that f (P (A)) ⊆ P (B) and f (A) ^
| f (P (A)) P (B). (Independence
here is calculated in B, i.e., in any e.c. model or universal domain in which B
is embedded.)
(iii) The free category of pairs, P, is the category whose objects are pairs and whose
morphisms are free embeddings.
Lemma 1.3. Assume that f : (A, P ) → (B, P ) is a free embedding. Then P (f (A)) =
f (P (A)) = f (A) ∩ P (B).
Proof. Clearly f (P (A)) ⊆ P (B) ∩ f (A) = P (f (A)), whereby
f (A) ^
| P (B) =⇒ P (f (A)) ^
| P (f (A)) =⇒ P (f (A)) ⊆ bdd(f (P (A)))
f (P (A))
f (P (A))
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ITAY BEN-YAACOV
But then P (f (A)) is a subset of:
bdd(f (P (A))) ∩ f (A) = f (bdd(P (A)) ∩ A) = f (P (A))
And the claim ensues.
qed1.3
It follows that if the inclusion (A, PA ) ⊆ (B, PB ) is a free embedding, then PB agrees
with PA on A, and it is legitimate to use P without further qualification.
Remark 1.4. One easily verifies that the identity is a free embedding, as well as the
composition of any two free embeddings f : A → B and g : B → C: g(f (P (A))) ⊆
g(P (B)) ⊆ P (C) and f (A) ^
| f (P (A)) P (B) =⇒ g(f (A)) ^
| g(f (P (A))) g(P (B)), so
g(B) ^
| g(P (B)) P (C) =⇒ g(f (A)) ^
| g(f (P (A))) P (C) by transitivity.
Since in a free embedding we have f (A) ∩ P (B) = f (P (A)), we may usually assume
that a free embedding is in fact an inclusion.
We aim to prove that P is a compact elementary category, as defined in [Ben03a].
Proposition 1.5. P is an abstract elementary category with amalgamation ([Ben03a,
Definition 2.27]).
Proof. Clearly, P is a concrete category; we verify the properties:
Injectiveness: Every free embedding is injective.
Tarski-Vaught property: Assume that we have free inclusions A ⊆ C and
B ⊆ C, such that A ⊆ B, and we need to show that the inclusion A ⊆ B is
free as well. It is clearly elementary, P (A) ⊆ P (B) and A ^
| P (A) P (C) =⇒
A^
| P (A) P (B).
Elementary chain property: Let (Ai ,S
P ) be pairs for i < λ, Ai ⊆ Aj freely
for every i ≤ j < λ, and set (B, P ) = i<λ (Ai , P ). By the finite character of
dividing Ai ^
| P (A ) P (B) =⇒ bdd(P (B) ∩ Ai ) = P (Ai ) for every i, so P (B) is
i
boundedly closed in B, and (B, P ) is a pair. Clearly Ai ⊆ B is a free extension
for every i, and (B, P ) is clearly minimal as such.
Amalgamation: Assume f : A → B and g : A → C are free. We may embed
(A, PA ), (B, PB ) and (C, PC ) in an appropriate universal domain of T such
that f and g be the identity maps and B ^
| A C. We know that both PB and
PC coincide with PA on A, but we still do not know that they coincide on
B ∩ C, so let us keep the distinction for a while. Define D = B ∪ C and
PD = PB ∪ PC . Then we have:
B^
| C =⇒ B ^
| PC =⇒ B ^
| PC =⇒ B ^
| PD
A
A
PA
PB
And similarly C ^
| P PD . Also, if a ∈ D ∩ bdd(PD ) then either a ∈ B or
C
a ∈ C. In the former case:
| a =⇒ a ∈ bdd(PB ) =⇒ a ∈ PB
B^
| PD =⇒ a ^
PB
PB
and in the latter a ∈ PC , so in either case a ∈ PD . This shows that D ∩
bdd(PD ) = PD , so (D, PD ) is a pair, and the inclusions B ⊆ D and C ⊆ D
are free. (It follows now by Lemma 1.3 that PA , PB and PC are simply PD
restricted to A, B and C, respectively.)
LOVELY PAIRS OF MODELS: THE NON FIRST ORDER CASE
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qed1.5
Therefore we have a notion of type: we recall that tp(A,P ) (a) = tp(B,P ) (b) if there are
free embeddings of (A, P ) and (B, P ) into some pair (C, P ) such that a and b have the
same image, and Sα (P) is the set (or class, as far as we know at this point) of types
of α-tuples in P.
The next step is to understand types:
Definition 1.6. Let (A, P ) be a pair, and a ∈ A a tuple.
(i) ac = Cb(a/P (A)) (calculated in T ).
As ac ∈ bdd(P (A)) ∩ dcl(aP (A)) ⊆ dcl(A), this definition takes place entirely
within A; and since the canonical base over P is invariant under free extensions,
we may write it rather as ac = Cb(a/P ) without concerning ourselves in which
specific pair this is taken.
(ii) â = a, ac .
(iii) The Morley class mcl(a) is the set of pure types of Morley sequences (of length
ω) in tp(a/ac ).
Lemma 1.7. If (A, P ) and a ∈ A are as above, then mcl(a) is the set of types of
Morley sequences in tp(a/P (A)).
Proof. Easy.
qed1.7
Lemma 1.8. Let (A, P ) and (B, P ) be two pairs, and a ∈ A, b ∈ B be two possibly
infinite tuples. Then the following are equivalent:
(i) tp(A,P ) (a) = tp(B,P ) (b) (in the sense of P).
(ii) mcl(A,P ) (a) = mcl(B,P ) (b)
(iii) mcl(A,P ) (a) ∩ mcl(B,P ) (b) 6= ∅
(iv) tpT (â(A,P ) ) = tpT (b̂(B,P ) ).
Proof.
(i) =⇒ (ii). mcl is invariant under free extensions.
(ii) =⇒ (iii). Morley sequences exist.
(iii) =⇒ (iv). A canonical base is in the definable closure of a Morley sequence.
(iv) =⇒ (i). We have â ∈ dcl(A), ac ∈ bdd(P (A)) and a ^
| ac P (A), so we may
consider (A, P ) as a free extension of (â, ac ). The same holds for (b̂, bc ) ⊆
(B, P ), and now apply amalgamation.
qed1.8
We need a tool that would tell us when two types belong to the same Morley class,
and this tool is the notion of concurrently indiscernible sequences. In fact, we prove
something a bit stronger than we actually need:
Definition 1.9. We say that sequences {(aji : j < α) : i < β} are concurrently
indiscernible over b if for every i < β and j0 < α the sequence (aji : j0 ≤ j < α) is
indiscernible over b ∪ {aji0 : j < j0 , i0 < β} (in other word, if every tail is indiscernible
over the union of all corresponding heads).
Notation 1.10. Let Sind
α (T ) ⊆ Sα×ω (T ) denote the set of types of indiscernible sequences of α-tuples. In particular, mcl(a) ⊆ Sind
|a| (T ).
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ITAY BEN-YAACOV
Lemma 1.11. Assume A = {ai : i < β} are tuples, not necessarily disjoint, in some
(e.c.) model of T , and qi ∈ Sind
|ai | (T ) for every i < β.
Then the following are equivalent:
(i) There is some pair (D, P ) where D ⊇ A and qi ∈ mcl(D,P ) (ai ) for every i.
(ii) There are concurrently indiscernible sequences (bji : j ≤ ω) with bωi = ai and
b<ω
² qi for every i.
i
Proof.
(i) =⇒ (ii). For every i < β, find a Morley sequence (bji : j ≤ ω) over P such
that bωi = ai and b<ω
² qi . Write B k = {bki : i < β}.
i
We now give a construction by induction on k < ω. At the beginning of the
kth step we assume that (bji : k ≤ j ≤ ω) is a Morley sequence over P B <k
for every i. During the step we may move (bji : k ≤ j < ω) around a bit in
order to obtain the same thing for k + 1 without moving AB <k , nor changing
<k
tp(b<ω
). From this point onward, B k is fixed as well.
i /ai P B
We may assume for every i that b<ω
| a P B <k A, whereby b<ω
| P B <k A. We
i
i
^
^
i
<ω
<k
may further assume that {bi : i < β} ∪ {A} is a P B -independent set. At
this point we fix B k = {bki : i < β} for the rest of the construction, and observe
that B k ^
| P B <k A.
We now work for each i separately: we observe that ai ^
| P B <k bk B k by the prei
vious paragraph and that (bji : k < j ≤ ω) is a Morley sequence over P B <k bki
with ai = bωi . Therefore there is an automorphism fixing ai P B <k bki that when
applied to (bji : k < j < ω) gives an P B ≤k -indiscernible sequence, and in fact
≤k
a Morley sequence over P B ≤k , as required. We now fix tp(b<ω
), and
i /ai P B
the construction continues.
At the end we obtain concurrently indiscernible Morley sequences over P with
the required types.
(ii) =⇒ (i). Let D = AB <ω and P (D) = D ∩ bdd(B <ω ). Then (D, P ) is a pair.
Since (bji : k ≤ j ≤ ω) is B <k -indiscernible, we have ai = bωi ^
| b[k,ω) B <k for
i
every k < ω, whereby ai ^
| b<ω P (D), and aci = Cb(bωi /b<ω
i ).
i
Since (bji : j ≤ ω) is an indiscernible sequence it is a Morley sequence over aci ,
(D,P )
and tp(b<ω
(ai ).
i ) ∈ mcl
qed1.11
Notation 1.12.
(i) For p ∈ S(P) define mcl(p) as mcl(a) for any a ² p: by
Lemma 1.8S this is well defined.
Similarly, for a set F ⊆ Sα (P), we define
S
mcl(F ) = p∈F mcl(p) = tpP (a)∈F mcl(a).
(ii) For tuples a<ω and b<ω (in an e.c. model of T ) such that all ai and bi are
of the same length α, say that a<ω =mcl b<ω if there exist aω = bω such that
(ai : i ≤ ω) and (bi : i ≤ ω) are concurrently indiscernible. Since T is thick,
this property is defined by a partial type rα (x<ω , y<ω ). We usually omit the
subscript α since it can be deduced from the context.
Then Lemma 1.11 gives:
Corollary 1.13.
(i) If q, q 0 ∈ Sα×ω (T ), then x<ω =mcl y<ω ∧ q(x<ω ) ∧ q 0 (y<ω ) is
consistent if and only if there is p ∈ tpα (P) such that q, q 0 ∈ mcl(p).
LOVELY PAIRS OF MODELS: THE NON FIRST ORDER CASE
9
(ii) Let p ∈ Sα (P) and q ∈ mcl(p). Then the partial type ∃y<ω [r(y<ω , x<ω ) ∧
q(y<ω )] defines the set mcl(p), which is in particular closed. (An existential
quantification on a partial type is equivalent to a partial type, by compactness.)
In particular, we may identify mcl(p) with the partial type ∃y<ω [r(x<ω , y<ω )∧q(y<ω )]
for any q ∈ mcl(p).
In addition if ai and bi are α-tuples for i < ω, then a<ω =mcl b<ω if and only if
0
a<ω =mcl b0<ω for every possible choice of corresponding sub-tuples a0i ⊆ ai , b0i ⊆ bi . It
follows that P-types satisfy the local character, namely the types of two infinite tuples
are equal if and only if the types of every two corresponding finite sub-tuples are equal.
We conclude that S(P) is a set type-space functor.
It is time now to define a language for P:
Definition 1.14. Let ϕ(x<k ) ∈ ∆, where each xi is an n-tuple. We define Rϕ as the
set of all p ∈ Sn (P) such that there is q(x<ω ) ∈ mcl(p) satisfying ϕ(x<k ) (that is to
say that mcl(p) is consistent with ϕ).
We interpret Rϕ as an n-ary predicate on pairs in the obvious way: if (A, P ) ∈ P and
a ∈ An then (A, P ) ² Rϕ (a) ⇐⇒ tp(A,P ) (a) ∈ Rϕ .
We define LP as the set of all such predicates, so |LP| = |L|. We also define ∆P =
∆0 (LP), that is the positive quantifier-free formulas in LP.
Remark 1.15. We cheat a bit, since Rϕ depends not only on ϕ but on the actual
decomposition of its free variables into k n-tuples, but we are just going to consider
that this information is contained in ϕ.
Ordinarily, the set of quantifier-free formulas is closed under conjunction, disjunction
and change of variables. We recall that if f : n → m is a map and ϕ(x<n ) a formula,
then ψ(y<m ) = ϕ(yf (0) , . . . , yf (n−1) ) is obtained from ϕ through a change of variables
by f , and we may also write ψ = f∗ (ϕ). However, in this particular language, the
finite disjunction and change of variables are not necessary:
Lemma 1.16.
(i) Let Rϕ (x<n ) be an n-ary predicate in this language, where
<n
<m
ϕ(x0 , . . . , x<n
be another tuple of variables and f : n → m
k−1 ) ∈ ∆. Let y
a map, and let us convene that by y f (<n) we mean the tuple y f (0) , . . . , y f (n−1) .
Then the formula f∗ (Rϕ )(y <m ) = Rϕ (y f (<n) ) is equivalent to Rψ (y <m ) where
f (<n)
f (<n)
<m
ψ(y0<m , . . . , yk−1
) = ϕ(y0
, . . . , yk−1 ).
(ii) Rϕ ∨ Rψ is equivalent to Rϕ∨ψ .
This means that every n-ary ∆P-formula is equivalent to a conjunction of Rϕ predicates, as finite disjunctions and changes of variables can be transferred to ϕ,
and similarly for partial ∆P-types.
Lemma 1.17.
(i) Let ρ(x<ω ) be a partial ∆-type, which
V we may assume to be
closed under finite conjunctions, and let Rρ (x) = ϕ(x<k )∈ρ Rϕ (x). Then p `
Rρ if and only if mcl(p) ∧ ρ is consistent.
(ii) Conversely, if Rϕ (x) ∈ LP, then mcl(RϕV
) is defined by the partial
mcl
type ∃y<ω = x<ω ϕ(y<k ); and if ρ(x) =
i<λ Rϕi (x) then mcl(ρ) =
V
).
mcl(R
ϕi
i<λ
10
ITAY BEN-YAACOV
Proof.
(i) Since mcl(p) is a closed set, we have that mcl(p) is consistent with ρ if
and only if mcl(p) is finitely consistent with ρ if and only if p ` Rρ .
(ii) Directly by Corollary 1.13
qed1.17
So let us see now what can be expressed in this language. All the following are easily
verifiable:
• Any complete P-type: for any p ∈ S(P) is defined by Rmcl(p) .
• Equality: x = y is defined by Rx<ω =y<ω (x, y).
• Existential quantification: if ρ(x, y) is a partial ∆P-type, and mcl(ρ) is defined
by ρ0 (x<ω , y<ω ), then ∃y ρ(x, y) is defined by R∃y<ω ρ0 . Therefore, our assumption that ∆ eliminates the existential quantifier (for T ) implies that so does
∆P (for P).
• Any ∆-formula ϕ(x): this is just Rϕ(x0 ) .
• x ∈ P : take Rx0 =x1 .
• Indiscernibility of sequences: write X = x<ω , and let ρ(X<ω ) say that (xj<ω :
j < ω) is an indiscernible sequence (which is possible since T is thick). Then
Rρ (X) says that X = (xj : j < ω) is an indiscernible sequence in the sense of
P. This shows that P is thick.
• Equality of types:
if T is semi-Hausdorff then ρ(x<ω , y<ω ) =
∃z<ω [x<ω =mcl z<ω ≡ y<ω ] is a partial type, and Rρ (x, y) defines the property
x ≡ y, so T P is semi-Hausdorff as well.
• If inequality is positive in T , we may say that x ∈
/ P , by Rx0 6=x1 (this can be
improved).
The last thing to prove is that this logic is compact.
Lemma 1.18. Let Σ(X) be some partial ∆P-type, where X is a possibly infinite tuple.
Then Σ is realised in P if and only if it is finitely realised in P.
V
Proof. Write mcl(Σ)(X<ω ) = x⊆X,ϕ(x)∈Σ mcl(ϕ)(x<ω ). Then Σ is realised if and only
if mcl(Σ) is consistent if and only if mcl(Σ) is finitely consistent if and only if Σ is
finitely realised.
qed1.18
And we conclude:
Definition 1.19. T P = ThΠP (P) is the negative universal theory of pairs in this
language.
Theorem 1.20. T P is a thick positive Robinson theory in ∆P, and S(P) = S(T P).
If T is semi-Hausdorff or Hausdorff, then so is T P.
If T is complete then so is T P; otherwise, there is a bijection between completions of
T and T P.
Proof. We showed that LP is a language for P which can define complete types and
satisfies weak compactness. Thus, by [Ben03a], T P is a positive Robinson theory in ΣP,
and S(P) = S(T P), where ΣP is the set of positive existential LP-formulas. However,
as we proved that the language ∆P eliminates the existential quantifier, we can replace
ΣP with ∆P.
We also already proved that T P is thick, and if T is semi-Hausdorff then so is T P.
LOVELY PAIRS OF MODELS: THE NON FIRST ORDER CASE
11
If T is Hausdorff, and p 6= p0 ∈ Sn (T P), then mcl(p) ∩ mcl(p0 ) = ∅, so they can be
separated by open sets. In other words, there are partial types ρ(x<ω ) and ρ0 (x<ω ),
inconsistent with mcl(p) and mcl(p0 ), respectively, such that ² ρ ∨ ρ0 . Then Rρ and
Rρ0 are inconsistent with p and p0 , respectively, and P ² Rρ ∨ Rρ0 , so p and p0 are also
separated by open sets.
If (A, P ) and (B, P ) are two pairs, and A and B embed in e.c. models of the same
completion of T , then we can amalgamate the two pairs over (∅, ∅). On the other
hand, if A and B belong to distinct completions of T then we cannot embed them
in a single e.c. model. Therefore the completions of T are in bijection with those of
T P.
qed1.20
Convention 1.21. We shall work in a universal domain U P for (a completion of) T P.
2. Lovely pairs
Since the origin of the theory of pairs is in lovely ones, we need to say something
about them.
Definition 2.1. Let κ > |T |. A pair (M, P ) is κ-lovely if:
(i) For every A ⊆ M such that |A| < κ, and for every type p ∈ S T (A), there is
a ² p in M with a ^
| A P (M ).
(ii) For every A ⊆ M with |A| < κ and every type p ∈ S(A) which does not divide
over P (A), there is a ² p in P (M ).
Definition 2.2. A set A in U P is free if A ^
| P (A) P .
This means that (A, P ) is freely embedded in the universal domain, so it determines
tpP(A).
Proposition 2.3. Let κ > |T |. Then a pair (M, P ) is a κ-saturated model of T P if
and only if it is κ-lovely.
Proof. Let (M, P ) be κ-saturated, and we want to prove that it is κ-lovely:
(i) Assume that A ⊆ M , |A| < κ, and a is some element possibly outside M . As
we are only interested in tpT (a/A), we may assume that a ^
| A M.
Set D = aM , P (D) = D ∩ bdd(P (M )). Then (D, P ) is a pair, and a free
extension of (M, P ). By saturation, there is an element a0 ∈ M such that
tpP(a/A) = tpP(a0 /A). Define b = (aA)c , and b0 = (a0 A)c , so:
b, b0 ∈ bdd(P (D)) = bdd(P (M )) ⊆ bdd(M )
(In fact, since M is |T |+ -saturated we have bdd(M ) = dcl(M ), but this is not
used here.) From tpP(a/A) = tpP(a0 /A) we obtain tpT (aAb) = tpT (a0 Ab0 ).
Then we have a0 A ^
| b0 P , but also:
a^
| M =⇒ a ^
| b =⇒ a0 ^
| b0 =⇒ a0 ^
| P,
A
A
A
A
as required.
(ii) Assume that A ⊆ M , |A| < κ, and a ^
| P (A) A. We may assume that
a^
| P (A) M so a ^
| P (M ) M . Let D = aM as above, but define P (D) =
12
ITAY BEN-YAACOV
D ∩ bdd(aP (M )). Then (D, P ) is a free extension of (M, P ), and tpP(a/A) is
realised in M .
For the converse, assume that (M, P ) is κ-lovely. Assume that A ⊆ M , |A| < κ and a
is an element of some free extension (N, P ) of (M, P ). Write µ = |A| + |T | < κ.
We may find B ⊆ P (M ) such that |B| ≤ µ < κ and A ^
| B P (M ). Replacing A with
A ∪ B we may assume that A is free. Now find C ⊆ P (N ) such that |C| ≤ µ < κ and
a^
| AC P (N ). Since A is free in M it is also free in N , so A ^
| P (A) C and therefore
0
0
0
there is C ⊆ P (M ) with C ≡A C. Then there is a ∈ M such that a0 C 0 ≡A aC and
a0 ^
| AC 0 P (M ). Then aCA and a0 C 0 A are both free sets with aCA ≡ a0 C 0 A, whereby
0
aCA ≡P a0 C 0 A, so in particular a ≡P
qed2.3
A a as required.
Corollary 2.4. Every pair (A, P ) has a free extension to a κ-lovely pair.
Proof. Just embed it freely in a sufficiently saturated model of T P.
qed2.4
Remark 2.5. Assume that T is complete, and consider the language LP = L ∪ {P }.
Then every two |T |+ -lovely pairs are elementarily equivalent in this language, and any
two free sets of cardinality ≤ |T | with the same LP -diagram have the same type (this
generalises results in [Poi83, BPV03]).
Indeed, since two such sets have the same P-type, they correspond by an infinite backand-forth in saturated structures. In particular, since the empty set is free, we have
the elementary equivalence.
However, this is just a special case of a more general observation: taking any two
saturated models of a cat (or in fact, any two equi-universal homogeneous structures),
and taking any relational language whose n-ary predicates are interpreted as subsets of
Sn (without any topological requirement), then they are elementarily equivalent in this
language. Of course, they have no reason to be saturated as models of their first-order
theory, and when they are not, this first-order theory is rather meaningless.
3. Independence in T P
3.1. Simplicity. We prove that T P is simple and characterise independence.
Proposition 3.1. The following conditions are equivalent for (possibly infinite) tuples
a, b, c in P:
(i) Whenever (ai bi ci : i < ω) ² mcl(abc), then b<ω ^
| a c<ω .
<ω
(ii) There exist (ai bi ci : i < ω) ² mcl(abc) such that b<ω ^
| a c<ω .
<ω
b | ac.
(iii) (abc)c ∈ bdd((ab)c , (ac)c ) and ab
b
^ â
b | ac.
(iv) b ^
| aP c and ab
b
^ â
(v) b ^
| aP c and (ab)c ^
| ac (ac)c .
Proof.
(i) =⇒ (ii). Clear.
(ii) =⇒ (iii). We are given (ai bi ci : i < ω) ² mcl(abc) such that b<ω ^
| a c<ω . This
<ω
c
is a Morley sequence in tp(abc/(abc) ), and we may assume that abc = a0 b0 c0 .
In particular, the sequence (ai : 0 < i < ω) is a Morley sequence over â,
LOVELY PAIRS OF MODELS: THE NON FIRST ORDER CASE
13
b whereby ab
b | a<ω , so:
indiscernible over ab,
^ â
b | ac
b | ac
b<ω ^
| c<ω =⇒ ab
b
^ b =⇒ ab ^
a<ω
a<ω
â
We also know that ab ^
| (ab)c a[1,ω) b[1,ω) , and that ac ^
| (ac)c (abc)c
ac ^
| (ab)c (ac)c (abc)c . We obtain:
=⇒
| c(abc)c
| a[1,ω) b[1,ω) c<ω =⇒ b ^
b<ω ^
| c<ω =⇒ b ^
c
c
a<ω
a(ab)
a(ab)
=⇒ abc
c
|
^
(abc)
(ab)c (ac)c
Since tp(abc/(abc)c ) does not divide over (ab)c , (ac)c , we obtain (abc)c =
Cb(abc/(abc)c ) ∈ bdd((ab)c , (ac)c ).
(iii) =⇒ (iv).
b | ac
ab
^ b =⇒ b
â
|
^
a(ab)c (ac)c
c =⇒ b ^
| c
c
a(abc)
(iv) =⇒ (v).
c
b | ac
| (ac)c
| (ac)c =⇒ (ab)c ^
ab
^ b =⇒ (ab) ^
â
ac
â
(v) =⇒ (i). We know that (ai bi ci : i < ω) is a Morley sequence over (abc)c , so
ai b i c i ^
| (abc)c a6=i b6=i c6=i for all i < ω. Then:
b ^
| c =⇒ b ^
| c(abc)c
c
c
a(abc)
a(ab)
ai b i c i ^
| a6=i b6=i c6=i =⇒ bi ^
| a6=i b<i c<ω =⇒ bi
c
c
(abc)
ai (ab)
|
^
b<i c<ω
a<ω (ab)c
By induction on i we obtain b<i ^
| a (ab)c c<ω for all i,
<ω
b<ω ^
| a (ab)c c<ω . Finally, (ab)c ^
| ac (ac)c gives us:
whereby
<ω
a<ω c<ω ^
| (ab)c =⇒ a<ω c<ω ^
| (ab)c =⇒ c<ω ^
| (ab)c
c
c
a
(ac)
a<ω
=⇒ b<ω ^
| c<ω
a<ω
As required.
qed3.1
Definition 3.2. If any of the equivalent conditions in Proposition 3.1 holds we say
that b ^
|Pa c.
Remark 3.3. Conditions (iv) and (v) of Proposition 3.1 were proposed independently, in
some form or another, by all three authors of [BPV03] as candidates for independence
in T P (in the case where T P is first order).
Theorem 3.4. T P is simple, and b ^
|Pa c if and only if tpP(b/ac) does not divide over
a.
14
ITAY BEN-YAACOV
Proof. We need to prove that ^
|P is an independence relation.
By Proposition 3.1, if abc ∈ U P and (ai bi ci : i < ω) ² mcl(abc), then b ^
|Pa c ⇐⇒
b<ω ^
| a c<ω . This gives immediate proofs for all the properties of an independence
<ω
relation, with the exception of the independence theorem for Lascar strong types, which
we treat separately.
|Pa bi for i < 2. Then in
So assume that lstpP(b0 /a) = lstpP(b1 /a), c0 ^
|Pa c1 , and ci ^
c0 ≡Ls ab
c1 , and we also have ab
ci | ac
particular ab
c0 ^
| â ac
c1 . By the
â
^ â ci for i < 2 and ac
c
independence theorem in T we can find b, d ^
| â ac
c0 ac
c1 such that b, d ≡ac
b
ci i , (abi ) .
As (ac0 c1 )c ∈ bdd((ac0 )c (ac1 )c ) we have in fact b, d ^
| â ac
[
| ac ac
[
0 c1 , so d ^
0 c1 =⇒
P
d^
| (ac c )c ac0 c1 . We may therefore realise d in P , and then realise b in U such that
0 1
b^
| a\
P =⇒ b ^
| ad c0 c1 P . In particular, ab ^
| d P (since d ∈ P and ac ∈ dcl(d)),
c0 c1 ,d
and d = (ab)c .
For i < 2, we get b ^
| a(ab)c ci P =⇒ abci ^
| (ab)c (ac )c P . Recall that (abi ci )c ∈
i
b ≡ac
c
bdd((abi )c (aci )c ), so abi ci ^
| (ab )c (ac )c P as well. This along with ab
ci abi yields
i
i
b ≡P
aci bi .
b | ac
We know that ab
| a(ab)c c0 c1 P =⇒ b ^
| aP c0 c1 , so b ^
|Pa c0 c1 , as re0 c1 , and b ^
^ â [
quired.
qed3.4
Notice that in the proof of the independence theorem, we used the assumption
c0 ≡Ls ab
c1 . This implies that:
lstpP(b0 /a) = lstpP(b1 /a) only to conclude that ab
â
Corollary 3.5. For every a ∈ U P, bddP(a) is P-interdefinable with bdd(â).
This still holds even if we consider hyperimaginary sorts of T P that are not inherited
from T .
Proof. One inclusion is clear. For the other, let bE ∈ bddP(a) be a hyperimaginary. We saw that tpP(b/ bdd(â)) is an amalgamation base, so it is equivalent to
tpP(b/ bddP(a)) and therefore implies tpP(b/bE ). Then every automorphism of U P
that fixes bdd(â) sends b to another realisation of tpP(b/bE ), and therefore fixes bE , so
bE ∈ dclP(bdd(â)).
qed3.5
Corollary 3.6. If T is supersimple, then so is T P.
Proof. Let a be a singleton and B = {bi : i < α} a set in U P. Let (aj , Bj : j < ω) ²
mcl(a, B) in U , and extend this to a similar 2ω-sequence (aj , Bj : j < 2ω). By supersimplicity, there are n < ω and I ⊆ α finite such that aω ^
| a ,b∈I
a<ω B<2ω .
<n
∈[0,n)∪[ω,2ω)
B[0,m)∪[ω,2ω) , and by removing
Then for every m ∈ [n, ω) we have aω ^
| a ,b∈I
<m [0,m)∪[ω,2ω)
the segment [m, ω) we obtain am ^
| a ,b∈I B<ω . On the other hand, increasing I some<m <ω
what, though keeping it finite, we may also assume that a<n ^
| b∈I B<ω . Combined with
<ω
the previous observations, an easy induction gives a<m ^
| b∈I B<ω for every m ∈ [n, ω),
<ω
whereby a<ω ^
| b∈I B<ω .
<ω
qed3.6
We conclude that a ^
|Pb∈I B, with |I| < ω, as required.
Remark 3.7. The approach we take here for the proof of simplicity and the characterisation of independence in T P is completely different than that which appears in
LOVELY PAIRS OF MODELS: THE NON FIRST ORDER CASE
15
[BPV03]. The basic improvement is in the equivalence b ^
|Pa c ⇐⇒ b<ω ^
| a c<ω which
<ω
does not appear there. Given this equivalence, all that is left to show is the independence theorem, which then gives us at once the simplicity of T P, the characterisation
of dividing, and the characterisation of Lascar strong types.
In fact, not knowing what hyperimaginary sorts in T P look like, the only way we know
how to prove that bddP(a) = dclP(bdd(â)) is through the independence theorem, so
might just as well obtain the other results at the same time.
Added in proof: Recent results suggest that the “obvious” definition of supersimplicity is too strong for general cats (more precisely, for those where the property
x 6= y is not positive). A better (and more permissive) definition appears in [Benb] for
Hausdorff cats. The analogue Corollary 3.6 for this definition is true, although it does
not seem possible to prove it solely with the tools introduced in [Benb].
3.2. Stability. We recall:
Definition 3.8.
(i) T is λ-stable if | Sn (A)| ≤ λ for every set |A| ≤ λ.
(ii) T is stable if it is λ-stable for some λ.
(iii) T is superstable if it is λ-stable for every λ ≥ 2|T | .
One can prove along the lines of the classical proof:
Fact 3.9. Let T be any cat.
(i) T is stable if and only if T is λ|T | -stable for every λ.
(ii) T is superstable if and only if it is stable and supersimple.
Theorem 3.10. If T is stable or superstable, then so is T P.
Proof. Assume that T is stable, and count P-types over a set A. Fix a sequence
(Ai : i < ω) ² mcl(A): for every a, tpP(a/A) is determined by tp(a<ω /A<ω ), for any
a<ω such that (ai Ai : i < ω) ² mcl(aA) (and such a<ω always exists). By stability:
P
T
|SλP(A)| ≤ |Sλ+ω
(A<ω )| ≤ (|A| + ω)λ+|T | , so |A| = µ|T | =⇒ |S|T
| (A)| = |A|.
The result of superstable follows from Fact 3.9 and Corollary 3.6.
qed3.10
3.3. One-basedness. We recall:
Definition 3.11. A simple cat T (not necessarily thick) is one-based if whenever (ai :
i < ω) is a Morley sequence in a complete Lascar strong type p then Cb(p) ∈ bdd(ai )
for some (every) i.
Lemma 3.12. A cat T is one-based if and only if, whenever (ai : i < ω) is an
indiscernible sequence, then (ai : 0 < i < ω) is independent over a0 .
Proof. Remember that every indiscernible sequence is a Morley sequence over some
set A: for example, a copy of the sequence. Setting c = Cb(ai /A), (ai ) is a Morley
sequence over c.
If T is one based, then we have c ∈ bdd(a0 ), so ai ^
| c a<i =⇒ ai ^
| a a[1,i−1] for every
0
i. Conversely, if (ai ) is a Morley sequence in some Lascar strong type p and c = Cb(p),
| a c =⇒
then c ∈ dcl(a≥2 ) so a1 ^
| c a0 =⇒ c = Cb(a1 /ca0 ) and a1 ^
| a a≥2 =⇒ a1 ^
0
0
c ∈ bdd(a0 ).
qed3.12
16
ITAY BEN-YAACOV
Proposition 3.13. If T is one-based then so is T P.
Proof. Let (aj : j < ω) be an indiscernible sequence in U P. Extend (aj : j < ω) to a
very long (aj : j < λ), and take (a<λ
: i < ω) ² mcl(a<λ ). Considering it rather as a
i
j
long sequence (a<ω : j < λ), we may extract an indiscernible sequence, which shows
that there are (a<ω
: i < ω) ² mcl(a<ω ) such that (aj<ω : j < ω) is indiscernible. Since
i
T is one-based, the sequence (aj<ω : 0 < j < ω) is a Morley sequence over a0<ω , whereby
(aj : 0 < j < ω) is a Morley sequence over a0 .
qed3.13
4. The description of T P in T and its functoriality
In the first section we constructed the abstract elementary category P, defined the
language ∆P, and proved that T P = ThΠP (P) is a positive Robinson theory in ∆P,
with S(T P) = S(P). In the topology on S(T P), closed sets are those defined by
partial ∆P-types, and equipped with this topology it is a compact type-space functor.
However, this topology could have been obtained more directly, using the categoric
point of view described in [Bena].
Recall that we defined Sind
α (T ) as the subset of Sα×ω (T ) which consists of types of
indiscernible sequence of α-tuples. If f : α → β is a map, and f×ω : α × ω → β × ω
∗
is its natural extension to ω-tuples, then f×ω
: Sβ×ω (T ) → Sα×ω (T ) restricts to a map
∗
ind
ind
ind
ind
: Sβ (T ) → Sα (T ), so S (T ) is a sub-functor of S×ω (T ).
f
P
Lemma 4.1.
(i) For every α there is a unique map dα : Sind
α (T ) → Sα (T ) satisfying dα (q) = p ⇐⇒ q ∈ mcl(p).
0
P
(ii) For closed sets F ⊆ Sind
α (T ) and F ⊆ Sα (T ), we have dα (F ) = RF and
−1
0
0
dα (F ) = mcl(F ), and these sets are closed. In particular, every dα is continuous and closed.
(iii) Let f : α → β be a map. Then the following diagram commutes, which makes
d : Sind (T ) → S(T P) a morphism of functors:
Sind
β (T )
f ind
dβ
∗
/ Sβ (P)
f∗
²
Sind
α (T )
dα
²
/ Sα (P)
ind
0
∗ −1
Moreover, if p ∈ Sα (T P), q ∈ d−1
(p) =
α (p) = mcl(p) ⊆ Sα (T ) and p ∈ f
ind
P
0
0
ind
f∗ (p) ⊆ Sβ (T ), then there is q ∈ mcl(p ) ∩ f∗ (mcl(p)) ⊆ Sβ (T ).
(iv) d : Sind (T ) → S(T P) is a quotient map, meaning that d is a surjective map,
and the topology on S(T P) is maximal such that d is continuous.
(i) For every q ∈ Sind
α (T ) there is at most one value that dα can take, since
p 6= p0 =⇒ mcl(p) ∩ mcl(p0 ) = ∅. Such a value always exists, as can be seen
by applying Lemma 1.11 with β = 1.
(ii) This is just what Lemma 1.17 says.
(iii) It is a fact that if a, b and P are given, then a sequence (ai : i < ω) is a Morley
sequence in tp(a/P ) if and only if there are b<ω such that (ai bi : i < ω) is a
Proof.
LOVELY PAIRS OF MODELS: THE NON FIRST ORDER CASE
17
Morley sequence in tp(ab/P ). Then commutativity is one direction, and the
moreover part is the other.
(iv) Each dα is surjective since mcl(p) 6= ∅ for every p ∈ Sα (P). A surjective,
closed and continuous map is a quotient map.
qed4.1
Thus, we could have defined the topology on S(P) from the beginning as the quotient
topology, without ever bothering to define a language explicitly. Then, the commutativity statement in Lemma 4.1.(iii) shows that f ∗ : Sβ (P) → Sβ (P) is continuous, and
the moreover part shows that f ∗ is closed. As every set mcl(p) is closed, the topology
on S(P) is T1 , and it is compact as the quotient of a compact topology.
In short, we could have skipped everything that comes after Corollary 1.13, and still
conclude that S(P) is a compact type-space functor, so there is a positive Robinson
theory T P in some language such that S(P) = S(T P), but this time also as topological
functors. In fact, we could have skipped the entire first section, constructing S(P)
directly as the quotient of Sind (T ) by the appropriate equivalence relation (but then,
of course, we wouldn’t know what it is that we are constructing).
This very abstract approach still seems (at least to the author) quite convenient,
and allows a few elegant observations. Recall from [Bena]:
Definition 4.2.
(i) Let αd be an ordinal and S, S 0 compact type-space functors.
Let d : S×αd 99K S 0 be a continuous partial map, meaning that (S×αd )I = SI×α ,
dom(d) ⊆ S×αd is a closed sub-functor, and d : dom(d) → S 0 is a continuous
surjective morphism of functors. If ϕ(x<n ) is a formula in the language of S 0
identify it with the closed set it defines ϕ ⊆ Sn0 , and let ϕ̃(x̄<n ) be the partial
type in the language of S defining d−1
n (ϕ) ⊆ Sn×αd .
Let x<n+m be a tuple of variables, and let:
^
ϕj (xij,0 , . . . , xij,kj −1 )
ψ(x<n ) = ∃x∈[n,n+m)
j<l
ψ̂(x̄<n ) = ∃x̄∈[n,n+m)
^
ϕ̃j (x̄ij,0 , . . . , x̄ij,kj −1 ),
j<l
where each ϕj is an kj -ary formula, and ij,s < m + n for j < l and s < kj .
Then (d, αd) is a description of S 0 in S, written d : S 99K S 0 , if whenever ψ, ψ̂
are as above and p ∈ dom(d) is in the right number of variables then:
(1)
p ` ψ̂ ⇐⇒ d(p) ` ψ
(ii) A description is closed if d is a closed map.
(iii) If T and T 0 are simple cats, then a morphism r : S(T ) → S(T 0 ) preserves
independence if whenever a, b, c and a0 , b0 , c0 are possibly infinite tuples in the
0
universal domains of T and T 0 , respectively, and tpT (a0 , b0 , c0 ) = r(tpT (a, b, c)),
0
then b ^
| Ta c ⇐⇒ b0 ^
| Ta0 c0 .
Lemma 4.3. Let qj (x<ω
<kj ) be partial types for j < l, each of which implying that
s
(x<kj : s < ω) is an indiscernible sequence of kj -tuples (in other words, qj ` dom(dkj )).
V
<ω
, . . . , yi<ω
) is consistent, where ij,t < n for every
Assume that π(y<n
) = j<l qj (yi<ω
j,0
j,k −1
j
18
ITAY BEN-YAACOV
j < l and t < kj . Then it can be realised in dom(d), that is to say that it has a
realisation which is an indiscernible sequence of n-tuples.
<ω
<λ
), and in addition for every j < l
) say that π(y<n
Proof. Fix a very big λ, and let π 0 (y<n
s
the sequence (yij,<k : s < λ) is indiscernible. Then π 0 is consistent by compactness,
j
s<ω
0
and let a<λ
<n ² π . By indiscernibility, we have a<n ² π for every increasing sequence
s0 < s1 < · · · < λ. As we took λ sufficiently big, we can extract an indiscernible
sequence (bs<n : s < ω) such that, for every t < ω there are st0 < · · · < stt−1 < λ such
st
<t
<ω
that b<t
<n ≡ a<n , whereby b<n ² π as required.
qed4.3
Theorem 4.4.
(i) The map d : Sind (T ) → S(T P), viewed as a partial map d :
S×ω (T ) 99K S(T P), is a closed description also noted d : T 99K T P, with a
factor αd = ω, and domain dom(d) = Sind (T ).
(ii) This description is functorial: if g : T → T 0 is any morphism of type-space
functors of thick simple cats, then there is a unique morphism g P : T P → T 0 P
that makes the following diagram commute:
TÂ
dT
g
Â
/ T0
Â
Âd
 T0
²
Â
²
TP
gP
/
T 0P
(iii) If g preserves independence, then so does g P.
Proof.
(i) Given Lemma 4.1, all that is left to prove is (1) of Definition 4.2.
⇐= follows from the moreover part of Lemma 4.1.(iii). =⇒ follows from
Lemma 4.3.
(ii) Just verify that if q, q 0 ∈ Sind
α (T ) belong to the same Morley class, then so do
0
g(q), g(q ).
(iii) This is immediate from Proposition 3.1.
qed4.4
We prove in [Bena] that a theory describable in a simple theory is simple. Thus, had
we taken the course proposed in the beginning of this section, we could have concluded
that T P is simple immediately, even without giving an explicit characterisation of
independence.
Recall also from [Bena]:
Definition 4.5. Let T be a simple cat and T 0 a stable one. Then a stable representation
of T in T 0 is a morphism r : S(T ) → S(T 0 ) satisfying the following additional condition
(called preservation of independence): If a, b, c are (possibly infinite) tuples in a model
0
of T , a0 , b0 , c0 in a model of T 0 and tpT (a0 , b0 , c0 ) = r(tpT (a, b, c)) then a ^
| b c ⇐⇒
0
0
a ^
| b0 c .
With a minor abuse of notation we may also write it as r : T → T 0 .
Corollary 4.6. Assume that T is simple and thick and has a thick stable representation
(that is r : T → T 0 where T 0 is stable and thick). Then so does T P.
Proof. Let r : T → T 0 be a stable representation. Then rP : T P → T 0 P preserves
independence and T 0 P is stable, so it is a stable representation.
qed4.6
LOVELY PAIRS OF MODELS: THE NON FIRST ORDER CASE
19
Lastly, we would like to relate the lovely pairs construction with another “standard”
construction, namely the addition of a generic automorphism.
Definition 4.7. For stable T , we let C A (T ) denote the category of boundedly closed
sets from T equipped with an automorphism σ.
If the category C A (T ) forms a simple cat in a language extending that of T , and whose
notion of independence is independence in T of σ-closures, then we denote this cat by
T A and say that T A exists.
By [Pil00], if T is first order then T A exists, and it can be further shown to be
Robinson. We do not wish to address here the issue of existence of T A in the general
case, so we will just assume that T A exists. We do know however from [Bena] that if
T A exists then we have a stable representation rA : T A → T×ω , which sends the type
of an element in T A to the type in T of its orbit under the automorphism.
Proposition 4.8. Assume that T A does exist as and is thick. Then (T P)A exists and
is equal to (T A )P, and we have a commutative diagram, where rA and rP
A are the stable
P
A
representations of T and TA , respectively:
TÂA
rA
Â
²
(T P)A = (T A )P
rP
A
/ T×ω
Â
Â
²
/ TP
×ω
Proof. Let C P(T A ) be the category of pairs in T A . Let (A, σ) ∈ C A (T P). Then
A is boundedly closed in the sense of T P, which means that Ac ⊆ dcl(P (A)) (i.e.,
A^
| P (A) P ), and both A and P (A) are boundedly closed in the sense of T . Writing it
as (A, σ, P ) it can also be viewed as a pair in the sense of T A and therefore an object
of C P(T A ).
This mapping from C A (T P) into C P(T A ) is a full and faithful functor: indeed, if
f : (A, P, σ) → (B, P, σ) is a mapping, then it is a morphism in the sense of either
category if and only if f (P (A)) = P (f (A)), f ◦ σA = σB ◦ f and f (A) ^
| f (P (A)) P (B)
(since σ is an automorphism of A, P (A) and P (B), independence in the sense of T and
of T A is the same). Moreover, this functor is co-final: every object of C P(T A ) embeds
into the image of an object of C A (T P).
This means that since C P(T A ) is an abstract elementary category so is C A (T P), and
they have the same type-spaces. Therefore these two categories are equivalent for our
purposes: we can use the language we chose for C P(T A ) also for C A (T P), and both
have the same positive Robinson theory in this language (T A )P = (T P)A . Finally
it is an easy exercise to see that independence in the sense of (T A )P coincides with
independence in the sense of T P of the σ-closures.
The commutativity of the diagram is also easy.
qed4.8
5. Lowness and negation
Definition 5.1.
(i) We say that a formula ϕ is clopen if it defines a clopen set
in the type-space. Equivalently, if ¬ϕ is equivalent to a positive formula (and
then we identify them).
20
ITAY BEN-YAACOV
(ii) We recall that a k-inconsistency
witness for a formula ϕ(x, y) is a formula
V
ψ(y<k ) such that ψ(y<k ) ∧ i<k ϕ(x, yi ) is inconsistent.
A formula ϕ(x, y) is low if it has a k-inconsistency witness ψ such that, for
every indiscernible sequence (ai ), {ϕ(x, ai )} is inconsistent if and only if ²
ψ(a0 , . . . , ak−1 ) (in other words, it has a universal inconsistency witness for
indiscernible sequences).
(iii) T is low if every formula is.
We recall that a cat T is Robinson if and only if the type-spaces are totally disconnected if and only if we can choose the language such that all basic formulas are
clopen. It is first order if and only if existential formulas are clopen as well.
Remark 5.2. If T is first order then ϕ(x, y) is low if and only if there is k < ω such that,
if (ai : i < ω) is indiscernible and {ϕ(x, ai )} is inconsistent then it is k-inconsistent.
The proofs of several of the results below can be simplified accordingly. However,
note
V
that being Robinson does not suffice, since we need the negation of ∃x i<k ϕ(x, yi ),
and this is an existential formula.
Notation 5.3. Let ϕ(x, y) be a T -formula. Then ϕ(P, y) = ∃x ∈ P ϕ(x, y) is positive.
If ψ is a k-inconsistency witness for ϕVthen ¬ψ ϕ(P, x) = Rψ (y) is positive as well.
If the existential formula ψ(ȳ) = ∃x i<k ϕ(x, yi ) defines a clopen set, then we write
¬k ϕ(P, y) = R¬ψ (y).
Lemma 5.4. If A ⊆ B and a ^
| A B then ϕ(x, a) divides over A if and only if it divides
over B.
Proof. A Morley sequence for a over B is also a Morley sequence over A.
qed5.4
Lemma 5.5. If a and b are tuples in U P satisfying precisely the same ϕ(P, y) formulas,
then a ≡P b.
Proof. Let A ⊆ P be such that a ^
| A P . Since b satisfied all ϕ(x, P ) predicates that
a does, then by compactness, we can find B ⊆ P such that aA ≡ bB. Assume now
that ² ϕ(c, b) for some c ∈ P . Then ² ϕ(P, b) =⇒² ϕ(P, a), whereby ϕ(x, a) does
not divide over P and therefore neither over A (since a ^
| A P ). Then ϕ(x, b) does not
qed5.5
divide over B either and b ^
| B P . This suffices to see that a ≡P b.
Lemma 5.6. Let a be a tuple in U P and ϕ(x, y) a T -formula. Then ϕ(x, a) does not
divide over P if and only if it is satisfied in P .
Proof. If ϕ(x, a) is realised in P , clearly it cannot divide over P . Conversely, assume
that it does not divide over P . Then there is a complete type p ∈ S(aP ) such that
p(x) ` ϕ(x, a) and p does not divide over P . By loveliness of the universal domain, we
can realise p in P .
qed5.6
W
Corollary 5.7. P ² ¬ϕ(P, y) ↔ ψ ¬ψ ϕ(P, y), where ψ varies over all inconsistency
witnesses for ϕ.
Proof. ϕ(x, a) is not satisfied in P if and only if it divides over P if and only if there is
a Morley sequence (ai : i < ω) for a over P such that {ϕ(x, ai )} is inconsistent if and
only if there is (ai : i < ω) ² mcl(a) satisfying an inconsistency witness for ϕ. qed5.7
LOVELY PAIRS OF MODELS: THE NON FIRST ORDER CASE
21
Corollary 5.8. If a and b are tuples in P, and b satisfies every formula of the form
ϕ(P, y) or ¬ψ ϕ(P, y) that a does, then a ≡P b.
Proof. In this case a and b satisfy precisely the same ϕ(P, y) formulas.
qed5.8
Lemma 5.9. A formula ϕ(x, y) is low if and only if for every λ there is a partial type
Φdiv
ϕ (y, Z), |Z| = λ, such that ϕ(x, a) divides over a set B of cardinality λ if and only
if ² Φdiv
ϕ (a, B).
Proof. Assume that ϕ is low, and let ψ be the universal inconsistency witness. Then
the partial type saying that there is a Z-indiscernible sequence (yi : i < ω) satisfying
y0 = y ∧ ψ(y0 , . . . , yk−1 ) will do.
For V
the converse, write the partial type saying that (yi : i ≤ ω) is indiscernible,
∃x i<k ϕ(x, yi ) for every k < ω, and ϕ(x, yω ) divides over y<ω . If this could be
realised, we could continue the sequence to length 2ω, in which case (yi : ω ≤ i < 2ω)
would be a Morley sequence over y<ω , whereby ϕ(x, yω ) cannot divide over y<ω . Then
this is inconsistent, so there are k0 < ω, ψ0 implied by the statement thatVthe sequence
is indiscernible, and ψ1 (ȳ) ∈ Φdiv
ϕ (yω , y<ω ), such that ψ0 (ȳ) ∧ ψ1 (ȳ) ∧ ∃x
i<k0 ϕ(x, yi )
is contradictory. Let k be the total number of yi appearing there, and write ψ0 ∧
ψ1 = ψ(y0 , . . . , yk−1 ). Then ψ is a k-inconsistency witness for ϕ, and we claim that
it is universal. Indeed, assume that (ai : i < ω) are indiscernible and {ϕ(x, ai )} is
inconsistent. Let aω continue this sequence, so ϕ(x, aω ) divides over a<ω : then ψ0
holds due to the indiscernibility, and ψ1 since ² Φdiv
ϕ (aω , a<ω ). This shows that ψ
witnesses that ϕ is low.
qed5.9
Remark 5.10. The converse part was first proved in a special case by Vassiliev.
Corollary 5.11. If ϕ is low, then ϕ(P, y) is clopen in T P. The converse holds if T is
Robinson.
Proof. For left to right, if ψ(ȳ) witnesses that ϕ(x, y) is low, then P ² ¬ψ ϕ(P, y) ↔
¬ϕ(P, y).
For the converse,
T assume that T is Robinson, and that all the formulas are clopen. Set
Φdiv
(y,
Z)
=
{tp(a, B) : ϕ(x, a) divides over B}, and it will be enough to show that
ϕ
div
² Φϕ (a, B) =⇒ ϕ(x, a) divides over B.
Assume then that ² Φdiv
ϕ (a, B), and set q(y, Z) = tp(a, B). For every formula χ we
have χ ∈ q if and only if ¬χ ∈
/ q if and only if there are a0 , B 0 such that ² χ(a0 , B 0 ) and
0
0
ϕ(x, a ) divides over B . We can realise B 0 in P and then realise a0 such that a0 ^
| B0 P .
0
0
Then ϕ(x, a ) divides over P , and ² ¬ϕ(P, a ).
This shows that ¬ϕ(P, y) ∧ Z ⊆ P ∧ q(y, Z) is finitely consistent. As ¬ϕ(P, y) is
positive, this is consistent, and we might just as well assume that it is realised by a, B.
But then ¬ϕ(P, a) =⇒ ϕ(x, a) divides over B.
qed5.11
Corollary 5.12. T is low if and only if every formula ϕ(P, y) is clopen. In this case
T is first-order, and the formulas ϕ(P, y), ¬ϕ(P, y) form a basis for the LP, so taking
them as basic formulas T P is Robinson.
Proof. If T is low then we know that every formula ϕ(P, y) is clopen. Conversely, we
know that P is a model of T , so if every formula ϕ(P, y) is clopen then T is first order
(existential formulas are clopen), and then we know that T is low.
qed5.12
22
ITAY BEN-YAACOV
Remark 5.13. If fact, when T is low with quantifier elimination, we can axiomatise
T P directly as a universal Robinson theory in the language L0 consisting of predicates
ϕ(P, y) for every formula ϕ(x, y) in the language of T :
For every n, m < ω and formulas ϕi (xi , yi ) for i < n and ψj (tj , zj ) for j < m, consider
the statement:
^
^
^
^
∀y<n z<m [ ϕi (P, yi ) ∧
¬ψj (P, zj )] → ∃x<n
ϕi (xi , yi ) ∧
Φdiv
ψj (zi , x<n )
i<n
j<m
i<n
j<m
V
Since
T is assumed to have quantifier elimination, the statement ∃x<n i<n ϕi (xi , yi )∧
V
div
j<m Φψj (zi , x<n ) is equivalent modulo T to a quantifier-free partial type. Therefore,
the statement above can be viewed as a universal theory in L0 . Take T 0 to be the
universal theory consisting of all universal L0 -sentences thus obtained. Then T 0 is a
Robinson theory, equivalent as a cat to T P (that is, has the same type-space).
This is proved in [BPV03].
We know that a stable theory is low if and only if it is first-order: one direction is
classical, the other was proved above. We can also prove:
Proposition 5.14. If T is stable and Robinson then so is T P.
Proof. Since T is stable, mcl(p) is a complete type for every p, whereby ¬Rϕ = R¬ϕ
for every formula ϕ, and T P is Robinson.
qed5.14
Question 5.15. Find a necessary and sufficient condition for T P to be Robinson.
References
[Bena]
[Benb]
[Ben03a]
[Ben03b]
[BPV03]
[HKP00]
[Kim98]
[KP97]
[Pil00]
[Poi83]
Itay Ben-Yaacov, Thickness, and a categoric view of type-space functors, Fundamenta Mathematicae (to appear).
, Uncountable dense categoricity in cats, preprint.
, Positive model theory and compact abstract theories, Journal of Mathematical Logic
3 (2003), no. 1, 85–118.
, Simplicity in compact abstract theories, Journal of Mathematical Logic 3 (2003),
no. 2, 163–191.
Itay Ben-Yaacov, Anand Pillay, and Evgueni Vassiliev, Lovely pairs of models, Annals of
Pure and Aplied Logic 122 (2003), 235–261.
Bradd Hart, Byunghan Kim, and Anand Pillay, Coordinatisation and canonical bases in
simple theories, Journal of Symbolic Logic 65 (2000), 293–309.
Byunghan Kim, Forking in simple unstable theories, Journal of the London Mathematical
Society 57 (1998), no. 2, 257–267.
Byunghan Kim and Anand Pillay, Simple theories, Annals of Pure and Applied Logic 88
(1997), 149–164.
Anand Pillay, Forking in the category of existentially closed structures, Connections between
Model Theory and Algebraic and Analytic Geometry (Angus Macintyre, ed.), Quaderni di
Matematica, vol. 6, University of Naples, 2000.
Bruno Poizat, Paires de structures stables, Journal of Symbolic Logic 48 (1983), no. 2,
239–249.
Itay Ben-Yaacov, Massachusetts Institute of Technology, Department of Mathematics, 77 Massachusetts Avenue, Room 2-101, Cambridge, MA 02139-4307, USA
E-mail address: [email protected]
URL: http://www-math.mit.edu/~pezz